Understanding Scale Factors in Similar Figures
When two figures are similar, their corresponding angles are equal, and their corresponding sides are in proportion. The ratio of any pair of corresponding lengths is called the length scale factor, denoted by k.
If one figure is an enlargement or reduction of another, every length in the first figure is k times the corresponding length in the second figure.
For example, if a triangle has sides 3 cm, 4 cm, and 5 cm, and a similar triangle has a length scale factor k = 2, then its sides are 6 cm, 8 cm, and 10 cm respectively.
- Length scale factor (k) = (length in image) ÷ (corresponding length in original)
- All corresponding lengths in similar figures have the same scale factor k
- To find an unknown length, multiply or divide by k
Relating Areas of Similar Figures
Areas of similar figures are related by the square of the length scale factor. If the length scale factor is k, then the ratio of their areas is k².
This is because area is a two-dimensional measurement, so both dimensions scale by k, resulting in an area scale factor of k × k = k².
For example, if a rectangle has an area of 20 cm² and a similar rectangle has a length scale factor of 3, then its area is 3² × 20 = 9 × 20 = 180 cm².
- Area scale factor = k²
- Area of image = k² × area of original
- To find unknown area, multiply or divide by k²
Relating Volumes of Similar Solids
Volumes of similar solids scale by the cube of the length scale factor. If the length scale factor is k, then the ratio of their volumes is k³.
Volume is a three-dimensional measurement, so all three dimensions scale by k, resulting in a volume scale factor of k × k × k = k³.
For example, if a cube has volume 8 cm³ and a similar cube has length scale factor 2, then its volume is 2³ × 8 = 8 × 8 = 64 cm³.
- Volume scale factor = k³
- Volume of image = k³ × volume of original
- To find unknown volume, multiply or divide by k³
Proving Similarity of Triangles Using Geometric Reasoning
Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.
There are three main criteria to prove triangle similarity: AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side).
AA: If two angles of one triangle are equal to two angles of another, the triangles are similar because the third angles must also be equal by angle sum property, ensuring all angles correspond and sides are proportional. SAS: If two sides of one triangle are in proportion to two sides of another triangle and the included angle between those sides is equal, the triangles are similar. SSS: If all three sides of one triangle are in proportion to the three sides of another triangle, the triangles are similar.
- Use angle equality and side ratios to establish similarity
- Apply AA, SAS, or SSS criteria
- Explain reasoning clearly when proving similarity
Solving Problems Involving Similarity
To solve problems involving similar figures, first identify the scale factor k by comparing corresponding lengths.
Use k to find unknown lengths by multiplying or dividing accordingly.
Calculate areas by multiplying or dividing by k², and volumes by k³, depending on the problem context and what is known or required to find. When proving similarity, clearly state which criterion (AA, SAS, or SSS) applies and justify why it holds based on given information or geometric properties. Always include units in your answers and check that your results are reasonable (e.g., areas and volumes increase or decrease consistently with scale factor).
- Identify length scale factor k from given data
- Calculate unknown lengths, areas, or volumes using k, k², or k³
- Use appropriate similarity criteria to prove triangles similar
- Explain each step logically and include units
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify scale factor | Find ratio of corresponding lengths | If side AB=5 cm and corresponding side DE=10 cm, k=10/5=2 | Always start by finding k to relate measurements |
| Calculate lengths | Multiply or divide by k | Length in image = k × length in original | Use k to find missing side lengths |
| Calculate areas | Multiply or divide by k² | Area image = k² × area original | Remember area scales with square of k |
| Calculate volumes | Multiply or divide by k³ | Volume image = k³ × volume original | Volume scales with cube of k |
| Prove triangle similarity | Use AA, SAS, or SSS criteria | Show two angles equal (AA) or sides proportional with included angle equal (SAS) | State and justify similarity criteria clearly |
Active recall
Quick check
1. If two similar triangles have a length scale factor of 3, what is the ratio of their areas?
The area scale factor is 3² = 9, so the ratio of their areas is 9:1.
2. A cube has edges of length 4 cm. A similar cube has edges of length 6 cm. What is the ratio of their volumes?
Length scale factor k = 6/4 = 1.5. Volume scale factor = k³ = 1.5³ = 3.375. So the volume ratio is 3.375:1.
3. How can you prove two triangles are similar using the SAS criterion?
Show that two pairs of corresponding sides are in proportion and the included angle between those sides is equal.
4. If the area of a shape is 50 cm² and the length scale factor to a similar shape is 0.5, what is the area of the similar shape?
Area scale factor = 0.5² = 0.25. Area of similar shape = 0.25 × 50 = 12.5 cm².